Draw arrows on curves tikz

Question

I have created a tikz plot for which I would now like to plot arrows on to show direction of flow. The current code used is:

\begin{tikzpicture}
        \draw[thick, ->] (0, -3) -- (0, 3) node [black, anchor=south] {$y$};
        \draw[thick, ->] (-3, 0) -- (3, 0) node [black, anchor=west] {$x$};
        \draw[scale=1,domain=0.33:3,smooth,variable=\x,black, thick] plot ({\x},{1/\x});
        \draw[scale=1,domain=-3:-0.33,smooth,variable=\x,black, thick] plot ({\x},{1/\x});
        \draw[scale=1,domain=0.33:3,smooth,variable=\x,black, thick] plot ({\x},{-1/\x});
        \draw[scale=1,domain=-3:-0.33,smooth,variable=\x,black, thick] plot ({\x},{-1/\x});
\end{tikzpicture}

Which produces the image

Now I want to add arrows onto each line separately. I require the direction of the arrow to be given by me (in the case I would like to use this in another example. They should occur at approximately half way through the curve. An example:

Answer 1

Something like this?

    \documentclass[border=3mm,
                   tikz,
                   preview
                   ]{standalone}
    \usetikzlibrary{decorations.markings,decorations.pathmorphing}
    \begin{document}
    \begin{tikzpicture}
\begin{scope}[decoration={markings,
    mark=at position 0.25 with {\arrow[blue,very thick]{>}},
    mark=at position 0.75 with {\arrowreversed[blue,very thick]{>}}}
              ]
\draw[postaction={decorate},thick,->] (0, -3) -- (0,3) node [black,above] {$y$};
\end{scope}
\begin{scope}[decoration={markings,
    mark=at position 0.25 with {\arrowreversed[blue,very thick]{>}},
    mark=at position 0.75 with {\arrow[blue,very thick]{>}}}
              ]
\draw[postaction={decorate},thick,->] (-3, 0) -- (3,0) node [black,right] {$x$};
\end{scope}
\begin{scope}[decoration={markings,
    mark=at position 22mm with {\arrow[red,very thick]{>}}}
              ]
\draw[postaction={decorate},thick] plot[domain=0.33:3]      (\x,{1/\x});
\draw[postaction={decorate},thick] plot[domain=-0.33:-3]    ({\x},{1/\x});
\draw[postaction={decorate},thick] plot[domain=0.33:3]      (\x,{-1/\x});
\draw[postaction={decorate},thick] plot[domain=-0.33:-3]    (\x,{-1/\x});
\end{scope}
    \end{tikzpicture}
    \end{document}

Edit: In the first try I didn't care for arrows direction yet nor add them to the coordinate axis. Now this is corrected. To obtain different direction of arrows, I define decorations in scope separate for each coordinate axis and separate for curves. For them I reverse domains for two of them, that the arrows have right direction.